This comment started as a short scribble ("I mostly agree", in essence) but I kept getting distracted, going away, coming back and adding more. It seems to have grown in length but not in focus.

I completely agree for cosmological spacetimes. I largely agree more generally, other than thinking one shouldn't be afraid of Landau-Lifshitz and company for fear of lack of "rigour", or even resorting to vectors[1] in appropriate circumstances. Everyone should already know doing so can be a footgun. If not, someone will call them out (reviewer 2 if unlucky).

That said, it's just a blog posting aimed at non-scientists. As I know you know, Carroll has well-regarded lecture notes and now a textbook [§3.5 in the lead up to eqn 3.92 is roughly the meat of the blog post, and for the sentence you picked out cf §3.8's discussion of the use of Killing's equation, around eqns 3.178 - 3.181, defining a total energy after slicing certain spacetimes (but not the cosmological one, of course)] for anyone who wants more "rigour". :-)

Another nice -- in fact, imho better, because of how it gets to eqn (3) [and the side comment in square brackets immediately after it -- it's not a gravitational statement!] -- blog entry covering similar territory is by Michael Weiss (the diagonalargument.com one) & John C Baez: https://math.ucr.edu/home/baez/physics/Relativity/GR/energy_... Cherry picking their support for the point you make: 'These pseudo-tensors have some rather strange properties. If you choose the "wrong" coordinates, they are non-zero even in flat empty spacetime. By another choice of coordinates, they can be made zero at any chosen point, even in a spacetime full of gravitational radiation. For these reasons, most physicists who work in general relativity do not believe the pseudo-tensors give a good local definition of energy density, although their integrals are sometimes useful as a measure of total energy.'

I will admit that your notion of "rigor" is unusual to see in strongly relativistic astrophysics (e.g. NS-NS GW 170817 literature), and practically unheard of in the low speed weak field limit. So what? Is using GEM's vectors there somehow wrong just because it's a pain to deal with relatively boosted frames?

I think you're wrong that 'Any way of defining "the energy of the whole universe" will involve what are called "pseudotensors"'. There are other approaches. More germane to astrophysics -- although this is perhaps arguing more with Weiss&Baez than you? -- there is lots of work on trying to define quasilocal energy without pseudotensors. See especially e.g. the first paragraph and endnote [1] of Brown & York 1993 PRD vol 47 iss 4 <https://harvest.aps.org/v2/journals/articles/10.1103/PhysRev...> (direct PDF). And now I should really make myself time to trawl through the recent relevant work of self-styled "mathematician who pretends to be a physicist", Steve McCormick <https://www.quasilocal.com/#section2>.

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[1] For fun: "Linear algebra is easy. Just remember [that] real numbers are vectors, polynomials are vectors, integrable functions are vectors, matrices are vectors, tensors are vectors, and so on. As I said, easy." @j_bertolotti lighting a fire https://twitter.com/j_bertolotti/status/1786407421755134325 And a golden reply, "All tensors are vectors. They're not in the same vector space as the vectors they're maps for, but a tensor product of vector spaces is another vector space, satisfying all the axioms." https://twitter.com/roystgnr/status/1786449834498511130